There is a mathematically rich theory about particle filters, and the central tool to make that go seems to be *Feynman-Kac formulae*.
The notoriously abstruse
Del Moral (2004);Doucet, Freitas, and Gordon (2001) are universally commended for unifying and making consistent the diffusion processes and Feynman-Kac formulae and “propagation of chaos” and their delicate relationships.
I will get around to them eventually, maybe?

Relate, apparently: Backward SDEs.

## References

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*Journal of Nonlinear Science*29 (4): 1563–1619.Cérou, F., P. Del Moral, T. Furon, and A. Guyader. 2011. “Sequential Monte Carlo for Rare Event Estimation.”

*Statistics and Computing*22 (3): 795–808.Chan-Wai-Nam, Quentin, Joseph Mikael, and Xavier Warin. 2019. “Machine Learning for Semi Linear PDEs.”

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*An Introduction to Sequential Monte Carlo*. Springer Series in Statistics. Springer International Publishing.Chuang, Howard Jen-Hao. 2010. “The Feynman-Kac Formula:Relationships Between Stochastic DifferentialEquations and Partial Differential Equations.” Honours, ANU.

Del Moral, Pierre. 2004.

*Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications*. 2004 edition. Latheronwheel, Caithness: Springer.Del Moral, Pierre, and Arnaud Doucet. 2010. “Interacting Markov Chain Monte Carlo Methods for Solving Nonlinear Measure-Valued Equations.”

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*Partial Differential Equations and Applications*2 (4): 48.Papanicolaou, Andrew. 2019. “Introduction to Stochastic Differential Equations (SDEs) for Finance.”

*arXiv:1504.05309 [Math, q-Fin]*, January.
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